- Email: [email protected]
Preliminary study of the interference of surface objects and rainfall in overland flow resistance
Catena 78 (2009) 154–158
Contents lists available at ScienceDirect
Catena j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c a t e n a
Preliminary study of the interference of surface objects and rainfall in overland flow resistance Gary Li ⁎ Department of Geography and Environmental Studies, California State University, East Bay, Hayward, CA 94542, USA
a r t i c l e
i n f o
Article history: Received 31 July 2008 Received in revised form 3 March 2009 Accepted 28 March 2009 Keywords: Overland flow Flow resistance Surface objects Rainfall Interference
a b s t r a c t Linear superposition approach, which states that composite resistance of different types of roughness elements equals to the sum of individual resistance, has been widely adopted in the study of overland flow hydraulics. The approach assumes that different roughness elements act independently in the flow. This seems to be implausible because roughness elements in shallow overland flow are often close to and interfering with each other. The interference, such as that between rainfall impact and stone generated vertex, may strengthen or restrain flow resistance on one another. To examine this possible interference in flow resistance, a set of flume experiments using surface objects and rainfall as roughness elements were conducted. Since the flow resistance due to each element cannot be measured directly, the linear superposition equation had to be rewritten using friction factor increment as variable. The experiments measured different friction factor increments in flows where Reynolds number was less than 2000 and Froude number less than 2. The result shows that the interference between rainfall and surface roughness is significant and the composite resistance does not statistically equal to the sum of individual resistance. The linear superposition approach is hence invalid in the case of rainfall and surface roughness elements in overland flow resistance. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
linear superposition approach) would be used in the determination of composite resistance:
Flow resistance is one of the most important aspects of overland flow study (Abrahams and Li, 1998; Smith et al., 2007) and is often defined by Darcy–Weisbach friction factor f which is determined by flow characteristics: 2
f = 8gds = u
ð1Þ
where g is acceleration due to gravity, d is mean flow depth, s is energy slope, and u is mean flow velocity. Different roughness elements, such as sand grains of the slope surface, stones and pebbles, and rainfall etc. generate different individual resistance. When multiple roughness elements are present in the flow, they form a composite resistance. It has been widely accepted that composite resistance of different types of roughness equals to the sum of individual resistance (Shen and Li, 1973; Rauws, 1988; Weltz et al., 1992; Parsons et al., 1994; Hu and Abrahams, 2005). In a case where surface objects (e.g. stone or pebbles) and rainfall are the roughness elements in consideration, the following additive equation (also called
⁎ Fax: +1 510 885 2353. E-mail address: [email protected]. 0341-8162/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.catena.2009.03.010
fR&I = fR + fI
ð2Þ
where fR is the resistance when surface objects is the only roughness element and its density is R, fI is the resistance when rainfall is the only roughness element and its intensity is I, fR&I is the composite resistance when both roughness elements, with surface object density being R and rainfall intensity being I, are present in the flow. Linear superposition approach was initially proposed for river flows (Einstein and Banks, 1950; Yen, 2002) where roughness elements, such as rainfall impact to river surface and stones at the river bed, are normally separated from each other by the depth of flow. It may not be appropriate to use in overland flow studies because roughness elements commonly interfere with each other in shallow flows. For example, surface objects in overland flow often generate disturbance and turbulent vortex which extend vertically to the entire flow profile (Bunte and Poesen, 1993). When rainfall is added to the flow, raindrop impacts directly interfere with turbulent vertex, either amplify or suppressing them, causing the composite resistance likely to differ from the sum of surface object resistance and rainfall resistance. To investigate the interference between surface objects and rainfall in overland flow resistance, a sound study would 1) measure flow
G. Li / Catena 78 (2009) 154–158
resistance due to surface objects fR and the flow resistance due to rainfall impact fI separately; and 2) measure the composite resistance due to the presence of both rainfall and surface objects fR&I; and then 3) compare the sum of fR and fI and measured fR&I. However, flow resistance measured in lab or in field is almost always a composite resistance, often composed of surface resistance and form resistance or surface resistance and rainfall resistance etc. Unfortunately, there is currently not a method able to measure individual resistance of surface object fR or individual resistance of rainfall fI directly, making it impossible to investigate their interference and to evaluate Eq. (2) in its current form. It is therefore necessary to introduce friction factors fR′, fI′, and fR′&I′ where the magnitudes of their corresponding roughness elements are R′, I′, and R′&I′, to re-write Eq. (2) as: fR&I −ðfR V + fI VÞ = fR − fR V + fI − fI V which in turn, considering fR′&I′ = fR′ + fI′, can be written as ΔfR&I = ΔfR + ΔfI
ð3Þ
Where ΔfR&I = fR&I − fR′&I′ is the increment of composite resistance when both surface object density and rainfall intensity increase from R&I to R′&I′; ΔfR = fR − fR′ is the increment of flow resistance when surface object density increases from R to R′. ΔfI = fI − fI′ is the increment of rainfall resistance when rainfall intensity increases from I to I′. It should be noted that the determination of ΔfR does not actually require surface object to be the only roughness element in the flow. When rainfall is present while its intensity remains constant, the increase of surface objects from density R to R′ also results in ΔfR. Similarly, the determination of ΔfI does not require rainfall to be the only roughness element in the flow. When surface object is present while its density remains constant, the increase of rainfall intensity from I to I′ also results in ΔfI. Now, linear superposition approach and the additive equation (Eq. (2)) can be examined through Eq. (3) if direct measurements of ΔfR&I, ΔfR, and ΔfI can be made. Another important aspect of this superposition approach is in its linearity. For Eq. (2) to hold, the coefficients of independent variables need to be constant. This means that the importance of each individual resistance has to remain unchanged whatever the magnitude of other individual resistance is. This seems dubious. Li and Abrahams (1999) have shown that the role of rainfall in flow hydraulics and sediment transport processes changes with increasing flow power. It has also long been known that wave resistance, as
155
another example, diminishes when the impact of other roughness elements increases (Rouse, 1938). The linearity of the additive equation can be assessed by statistical regression as long as the experiment provides data that covers a wide range of flow conditions and diversified density of roughness elements. This study examines the validity of linear superposition approach when rainfall and surface roughness are considered in overland flow resistance. 2. Experimental setup and methods To test Eq. (3), a set of experiments were conducted in a 5.2 m long and 0.4 m wide flume with a smooth aluminum bed and Plexiglas walls (Fig. 1). It consisted of two parts: a lower part 3.6 m long and an upper steeper part 1.6 m long. For the experiments the lower part of the flume was inclined at a slope s of 2.0°. Water entered the upper part of the flume by overflowing from a head tank. The water inflow of 44.9 to 170.4 cm3 s− 1 was controlled by a gate valve and measured with a rotameter. Water discharge Q at the end of the flume was calculated by adding the rainfall onto the flume to the water inflow from the head tank. The density of the water ρ was assumed to be 1000 kg m− 3, while the kinematic viscosity ν was determined from water temperature which was 17 °C. The study focuses on two roughness elements: simulated rainfall and surface objects. PVC pipe connectors of 3.3 cm diameter, filled with concrete and sealed at both ends, were used as surface objects. They were randomly placed in the flume, perpendicular to the flume bed, and were tall enough to fully protrude above the water surface. The proportion of the bed covered by connectors R ranged from 0 to 20%. Rainfall was simulated using Spraco–Lechler full cone jet nozzles (Luk et al., 1986) mounted 0.3 m apart and 3.6 m above the center of the flume. Rainfall intensity I in experiment ranged from 0, 1.22, to 1.84 mm min− 1 (see the detailed experiment design in Table 1). The median drop sizes (Laws and Parsons, 1943) associated with these two intensities were 2.0 and 2.4 mm, respectively. Drop size increased with rainfall intensity because the spray cones intersected when more than one nozzles were operating, causing the water drops to collide and form larger drops. Given that the flow velocity from each nozzle exceeded 5 m s− 1, most drops reached terminal velocity. The spatial uniformity of the rainfall was evaluated using eight rain gages during nine 30-minute events. The coefficient of variation averaged 0.093, indicating a highly uniform distribution. The flow velocity u was determined at a 1.6 m long section of the lower end of the flume by a dye tracing method. The experiments were conducted in clear water flow with Reynolds number Re b 2000
Fig. 1. Flume set up.
156
G. Li / Catena 78 (2009) 154–158
Table 1 Experiment design. Surface object density R (%)
Rainfall intensity I (mm min− 1)
Measured friction factor f
14 (R2) 20 (R3) 7 (R1) 14 (R2) 20 (R3) 7 (R1) 14 (R2) 20 (R3)
0 (I0) 0 (I0) 1.22 (I1) 1.22 (I1) 1.22 (I1) 1.84 (I2) 1.84 (I2) 1.84 (I2)
fR2I0 fR3I0 fR1I1 fR2I1 fR3I1 fR1I2 fR2I2 fR3I2
(Re = 4q/ν, where q is unit flow discharge). The mean velocity was then obtained by multiplying a constant 0.67 to leading edge velocity (Horton et al., 1934). For each flow velocity, measurements were repeated 3 times from which two closest results were averaged to yield the mean value u. Mean flow depth d was then calculated using d = Q/uw and w = W(1 − R), where w is the mean flow width, W is the flume width, and R is the roughness element density which is determined from the total area covered by PVC pipe connectors divided by flume surface area. In total, 40 experiments were conducted and the details are listed in Table 2. Decades of study of flow hydraulics has shown that f varies with Re. To obtain Δf it is therefore essential to compare f values at the same Re. In current experiments, f was measured at slightly different Re (see Table 2) because it is impossible to precisely control the flow
discharge Q (and therefore the unit flow discharge q from which Re was determined) with a mechanic gate valve. To tackle the issue, the data was organized into five groups for which data fitting was performed to align comparable f to the same Re (see Fig. 2a–e). The method used straight lines to connect dots pair by pair on f−Re chart and then linearly interpolated f values to a given Re. The location of lines regarding Re is so chosen that only interpolation (no extrapolation) is involved. Δf of different roughness densities or intensities was then determined by comparing f values at given Re (see Table 3). Take Group a. as an example, ΔfR&I was calculated by comparing flow resistance fR2&I2 and fR1&I1when both surface object density and rainfall intensity increased from R1 (7%) and I1 (1.22 mm min− 1) to R2 (14%) and I2 (1.84 mm min− 1). ΔfR was determined by comparing flow resistance fR2&I1 and fR1&I1when surface object density increased from R1 to R2 while the rainfall intensity remained unchanged as I1. ΔfI was determined by comparing flow resistance fR1&I2 and fR1&I1 when rainfall intensity increased from I1 to I2 while the surface object density remained unchanged as R1. The comparison results are listed in Table 4. 3. Results Corresponding to the rationales stated in the Introduction, the experiment results were analyzed in two ways: (i) direct comparison of Δf to examine the validity of Eq. (3); and (ii) multiple regression of all involved resistance to investigate the possible variation of coefficients for each friction factor.
Table 2 Experiment results. Experiment number
Reynolds number Re
Froude number Fr
Base flow discharge Q (cm3 s− 1)
Roughness density R (%)
Rain intensity I (mm min− 1)
Mean flow velocity U (cm s− 1)
Friction factor f
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1084 1287 1490 1695 1899 1020 1220 1422 1626 1830 1008 1211 1416 1620 1824 1091 1292 1496 1700 1904 1022 1221 1423 1626 1830 1008 1211 1414 1619 1823 1098 1298 1501 1705 1909 1025 1223 1423 1626 1829
1.4 1.8 1.8 1.8 1.8 1.3 1.6 1.6 1.7 1.7 1.4 1.5 1.7 1.7 1.8 1.4 1.4 1.4 1.5 1.6 1.1 1.2 1.3 1.3 1.3 1.3 1.4 1.5 1.6 1.5 1.1 1.2 1.2 1.3 1.3 0.9 1.1 1.1 1.2 1.2
80.3 100.7 121.3 141.9 162.5 58.9 79.1 99.5 120.1 140.7 94.2 113.2 132.2 151.3 170.4 72.6 91.4 110.4 129.5 148.6 51.3 69.9 88.7 107.7 126.8 87.6 105.2 122.9 140.7 158.4 66.1 83.5 101.1 118.8 136.6 44.9 62.1 79.5 97.1 114.8
7 7 7 7 7 7 7 7 7 7 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84 0 0 0 0 0 1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84 0 0 0 0 0 1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84
18.2 22.1 23.3 24.6 25.2 16.6 20.4 21.2 22.8 24.4 17.7 19.1 22.1 23.1 24.6 17.4 19.1 20.0 21.9 23.3 14.5 16.9 18.2 19.9 20.6 16.5 18.3 20.2 22.1 22.6 15.1 16.9 18.3 19.7 20.8 13.0 15.3 17.0 18.2 18.6
3.50 2.31 2.28 2.19 2.29 4.30 2.78 2.88 2.65 2.45 3.51 3.34 2.54 2.56 2.36 3.99 3.57 3.60 3.14 2.91 6.51 4.92 4.59 4.02 4.05 4.35 3.81 3.31 2.90 3.07 6.18 5.23 4.72 4.34 4.10 9.05 6.60 5.60 5.25 5.47
G. Li / Catena 78 (2009) 154–158
157
Fig. 2. Illustrations of linear data fitting. Vertical axes represent the Darcy–Weisbach friction factor. Horizontal axes are Reynolds numbers. Dashed lines indicate the Reynolds numbers at which friction factor f is interpolated.
3.1. Direct comparison of Δf Five groups of a total of 20 comparisons between composite resistance and the sum of individual resistance were made. Values in columns ΔfR&I and ΔfR + ΔfI in Table 4 clearly demonstrated that ΔfR&I ≠ ΔfR + ΔfI :
ð4Þ
Simple t-test confirmed that ΔfR&I is significantly larger than ΔfR + ΔfI for all 16 comparisons in Groups a. to d. at 95% confidence level. The difference in percentage, indicated by ΔfR&I − (ΔfR + ΔfI) / (ΔfR + ΔfI), varies from 8% to 210%, averaging 47%. For Group e. result, t-test showed that there was no significant difference between ΔfR&I and ΔfR + ΔfI. Two of the four comparisons in Group e. showed that ΔfR&I was slightly less than ΔfR + ΔfI (the difference is −3% and −8%). Further examination of the Fig. 2 revealed that the result of Group e. was mainly influenced by one set of experiments where R = 14% and I = 1.22 mm Table 3 Determination of friction factor increments. Comparison group
ΔfR&I
ΔfR
ΔfI
a b c d e
fR2I2 − fR1I1 fR3I1 − fR2I0 fR3I2 − fR1I1 fR3I2 − fR2I0 fR3I2 − fR2I1
fR2I1 − fR1I1 fR3I0 − fR2I0 fR3I1 − fR1I1 fR3I0 − fR2I0 fR3I1 − fR2I1
fR1I2 − fR1I1 fR2I1 − fR2I0 fR1I2 − fR1I1 fR2I2 − fR2I0 fR2I2 − fR2I1
Note: ΔfR&I is the increment of composite resistance when both surface object density and rainfall intensity increase from R&I to R′&I′. ΔfR is the increment of surface object resistance when surface object density increases from R to R′ while rainfall intensity remains the same. ΔfI is the increment of rainfall resistance when rainfall intensity increases from I to I′ while surface object density remains the same.
min− 1. The measured fR2I1 in the experiments seemed to be lower than it should be (see Fig. 2b. and e.). This was likely due to a specific arrangement of surface objects in the flume (which was done randomly)
Table 4 Results of friction factor increments and their comparison. Reynolds number
ΔfR&I
ΔfR
Group a. 1091 1292 1496 1830 Group b. 1098 1298 1501 1823 Group c. 1098 1298 1501 1829 Group d. 1025 1223 1423 1823 Group e. 1098 1298 1501 1829
fR2I2 − fR1I1 2.48 2.49 2.10 1.79 fR3I1 − fR2I0 2.74 2.23 2.17 1.84 fR3I2 − fR1I1 4.73 3.92 3.19 3.21 fR3I2 − fR2I0 5.55 3.30 3.07 3.09 fR3I2 − fR2I1 4.14 2.64 1.86 2.47
fR2I1 − 0.52 1.26 1.32 0.73 fR3I0 − 0.68 0.60 0.60 0.71 fR3I1 − 2.76 2.93 2.44 1.93 fR3I0 − 0.81 0.47 0.83 0.71 fR3I1 − 2.20 1.66 1.13 1.20
ΔfI fR1I1
fR2I0
fR1I1
fR2I0
fR2I1
fR1I2 − 0.28 0.51 0.52 0.19 fR2I1 − 0.54 0.58 1.04 0.65 fR1I2 − 0.29 0.51 0.51 0.19 fR2I2 − 2.99 1.61 1.56 1.68 fR2I2 − 1.91 1.22 0.79 1.05
ΔfR + ΔfI
ΔfR&I −ðΔfR + ΔfI Þ ðΔfR + ΔfI Þ
0.80 1.77 1.84 0.92
210% 41% 14% 95%
1.22 1.18 1.64 1.36
125% 89% 32% 35%
3.05 3.44 2.95 2.12
55% 14% 8% 51%
3.8 2.08 2.39 2.39
46% 59% 28% 29%
4.11 2.88 1.92 2.25
1% − 8% − 3% 10%
fR1I1
fR2I0
fR1I1
fR2I0
fR2I1
Note: Detailed flow condition of specific f, denoted by its subscript, can be found in Table 1. The Reynolds number in each group corresponds to dashed lines in Fig. 2.
158
G. Li / Catena 78 (2009) 154–158
that happened to assemble a less resistance and promoted a faster water flow. 3.2. Multiple regression analysis To investigate whether the role of individual resistance changes across the range of experiment conditions, a multiple regression was performed with f as dependent variable, R, I, and the product of both R ⁎ I as well as Re as independent variables. Initial regression yielded: 0:873
or
− 0:867
f = ð273:4R + 758:3I + 98:7RTIÞ Re 0:873 − 0:867 f = ð273:4R + ð758:3 + 98:7RÞI Þ Re
ð5Þ
with residual corrected R2 = 0.934 and N = 40. The result confirmed, as expected, that f increases with R and I and decreases with Re. It seems that f also increases with the term R ⁎ I. If so, the coefficient for independent variable I would be (758.3 + 98.7R) which is not a constant but varies with the value of R (the same is true for the coefficient of independent variable R). This would have concluded that the contribution of I to f depends on R (or vice versa). However, a significance test showed that the inclusion of the term R ⁎ I was statistically insignificant and improved the R2 only by 0.9%. Power coefficient for Re (− 0.867) was not statistically different from − 1 either and improved the R2 merely by 0.6%. This agreed with Moody diagram in which f varies with Re− 1 for Re b 2000. A revised regression without R ⁎ I and with power coefficient for Re as − 1 yielded: 1:204
f = ð59:0R + 343:8IÞ
Re
−1
ð6Þ
2
with residual corrected R = 0.916 and N = 40. Significance test showed that R, I and Re were all statistically significant. 4. Conclusion and discussion The current experiments were conducted in a smooth-bed flume of 2° slope. Reynolds number Re was less than 2000. Two roughness elements were investigated for their possible interference in overland flow resistance: surface objects and rainfall. The result shows that surface objects (and their turbulent vortex) interfere with raindrop impact to the flow which makes the composite resistance differ to the sum of the two individual resistances. The interference may restrain or strengthen friction factor on one another, leading the composite resistance to decrease or to increase. For river flows, Vanoni and Nomicos (1959) had published a similar result which showed that composite resistance decreased by about 5 to 28% when sediment load was introduced to turbulent channel flow. They ascribed the decreasing resistance to sediment load suppressing flow turbulence. Current experiment also tried to explore whether the interference of individual resistance increases or decreases composite resistance. The analysis did show that in most cases ΔfR&I N ΔfR + ΔfI. Great care must be exercised here not to conclude that fR&I N fR + fI though. It would require establishment of other conditions that are unknown at this point (e.g. the linear relationship between fR&I and R&I, fR and R, fI and I). Further study with extended flow condition and diversified roughness elements is needed to reveal the full spectrum of the relationship in this matter. Other studies suggested that the contribution of each roughness element to flow resistance may not be independent of other roughness elements (e.g. rough elements generate wakes that reduce grain resistance, Morris, 1955; Abrahams and Parsons, 1991) and the coefficients of independent friction factors in the additive equation may not be constant (e.g. rainfall resistance declines with Re, Shen and Li, 1973). However, current experiment and analysis did not reveal the changing nature of rainfall in its contribution to flow resistance with varying density of surface objects (or vice versa). This was most
likely due to a relatively narrow range of roughness density covered in the experiments. It is suggested to further investigate in the future the possible diminishing role of a roughness element (e.g. rainfall) in overland flow resistance due to increased magnitude of other roughness element (e.g. surface objects). Finally, it is worth pointing out that roll waves can easily develop in flume and in field experiments (Benjamin, 1957; Yoon and Wenzel, 1971; Emmett, 1978;) especially when flow is barely impacted by roughness elements (Rouse, 1938) or when Froude number Fr, defined as u/(gd)0.5, is greater than 2 (Brock, 1970; Liggett, 1975; Liu et al., 2005). Roll wave is a form of energy expenditure and adds another resistance to the flow. In order not to complicate the analysis, the current study did not include the experiments with visually identifiable roll waves. Acknowledgements The author greatly appreciates the support of Professor Sean Bennett, Department of Geography, State University of New York at Buffalo and Professor Shixiong Hu, Department of Geography, East Stroudbury University of Pennsylvania for their support and assistance during the experiment conducted in the Geomorphology Lab at State University of New York at Buffalo. The author also thanks Professor Michael Woldenberg and Professor Athol Abrahams, Department of Geography, State University of New York at Buffalo for their valuable comments and suggestions. References Abrahams, A.D., Parsons, A.J., 1991. Resistance to overland flow on desert pavement and its implications for sediment transport modeling. Water Resour. Res. 27, 1827–1836. Abrahams, A.D., Li, G., 1998. Effect of saltating sediment on flow resistance and bed roughness in overland flow. Earth Surf. Process. Landf. 23, 953–960. Benjamin, T.B., 1957. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574. Brock, R.R., 1970. Periodic permanent roll waves. J. Hydraul. Div. Am. Soc. Civ. Eng. 95, 1401–1428. Bunte, K., Poesen, J., 1993. Effects of rock fragment covers on erosion and transport of noncohesive sediment by shallow overland flow. Water Resour. Res. 29, 1415–1424. Einstein, H.A., Banks, R.B., 1950. Fluid resistance of composite roughness. Trans. Am. Geophys. Union 31 (4), 603–610. Emmett, W.W., 1978. Overland flow. In: Kirkby, M.J. (Ed.), Hillslope Hydrology. Wilely, New York, pp. 145–175. Horton, R.E., Leach, H.R., Van Vliet, R., 1934. Laminar sheet-flow. Trans. Am. Geophys. Union 15, 393–404. Hu, S., Abrahams, A.D., 2005. The effect of bed mobility on resistance to overland flow. Earth Surf. Process. Landf. 30, 1461–1470. Laws, J.O., Parsons, D.A., 1943. The relation of raindrop size to intensity. Trans. Am. Geophys. Union 24, 452–460. Li, G., Abrahams, A.D., 1999. Controls of sediment transport capacity in laminar interrill flow on stone-covered surfaces. Water Resour. Res. 35, 305–310. Liggett, J.A., 1975. Unsteady Flow in Open Channels. Water Resources Publications, Fort Collins, Colorado. Liu, Q.Q., Chen, L., Li, J.C., Singh, V.P., 2005. Roll waves in overland flow. J. Hydrol. Eng. 10 (2), 110–117. Luk, S.H., Abrahams, A.D., Parsons, A.J., 1986. A simple rainfall simulator and trickle system for hydro-geomorphological experiments. Phys. Geogr. 7, 344–356. Morris, H.M., 1955. Flow in rough conduits. Trans. Am. Soc. Civ. Eng. 120, 373–410. Parsons, A.J., Abrahams, A.D., Wainwright, J., 1994. On determining resistance to interrill overland flow. Water Resour. Res. 30, 3515–3521. Rauws, G., 1988. Laboratory experiments on resistance to overland flow due to composite roughness. J. Hydrol. 103, 37–52. Rouse, H., 1938. Fluid Mechanics for Hydraulic Engineers. McGraw-Hill, New York. Shen, H.W., Li, R.-M., 1973. Rainfall effect on sheet flow over smooth surface. J. Hydraul. Div. Am. Soc. Civ. Eng. 99, 771–792. Smith, M.W., Cox, N.J., Bracken, L.J., 2007. Applying flow resistance equations to overland flows. Prog. Phys. Geogr. 31 (4), 363–387. Vanoni, V.A., Nomicos, G., 1959. Resistance properties of sediment-laden streams. J. Hydraul. Div. Am. Soc. Civ. Eng. 85 (5), 77–107. Weltz, M.A., Arslan, A.B., Lane, L.J., 1992. Hydraulic roughness coefficient for native rangelands. J. Irrig. Drain. Eng. 118 (5), 776–790. Yen, B.C., 2002. Open channel flow resistance. J. Hydraul. Eng. 128 (1), 20–39. Yoon, Y.N., Wenzel Jr., H.G., 1971. Mechanics of sheet flow under simulated rainfall. J. Hydraul. Div. Am. Soc. Civ. Eng. 97, 1367–1386.
Contents lists available at ScienceDirect
Catena j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c a t e n a
Preliminary study of the interference of surface objects and rainfall in overland flow resistance Gary Li ⁎ Department of Geography and Environmental Studies, California State University, East Bay, Hayward, CA 94542, USA
a r t i c l e
i n f o
Article history: Received 31 July 2008 Received in revised form 3 March 2009 Accepted 28 March 2009 Keywords: Overland flow Flow resistance Surface objects Rainfall Interference
a b s t r a c t Linear superposition approach, which states that composite resistance of different types of roughness elements equals to the sum of individual resistance, has been widely adopted in the study of overland flow hydraulics. The approach assumes that different roughness elements act independently in the flow. This seems to be implausible because roughness elements in shallow overland flow are often close to and interfering with each other. The interference, such as that between rainfall impact and stone generated vertex, may strengthen or restrain flow resistance on one another. To examine this possible interference in flow resistance, a set of flume experiments using surface objects and rainfall as roughness elements were conducted. Since the flow resistance due to each element cannot be measured directly, the linear superposition equation had to be rewritten using friction factor increment as variable. The experiments measured different friction factor increments in flows where Reynolds number was less than 2000 and Froude number less than 2. The result shows that the interference between rainfall and surface roughness is significant and the composite resistance does not statistically equal to the sum of individual resistance. The linear superposition approach is hence invalid in the case of rainfall and surface roughness elements in overland flow resistance. © 2009 Elsevier B.V. All rights reserved.
1. Introduction
linear superposition approach) would be used in the determination of composite resistance:
Flow resistance is one of the most important aspects of overland flow study (Abrahams and Li, 1998; Smith et al., 2007) and is often defined by Darcy–Weisbach friction factor f which is determined by flow characteristics: 2
f = 8gds = u
ð1Þ
where g is acceleration due to gravity, d is mean flow depth, s is energy slope, and u is mean flow velocity. Different roughness elements, such as sand grains of the slope surface, stones and pebbles, and rainfall etc. generate different individual resistance. When multiple roughness elements are present in the flow, they form a composite resistance. It has been widely accepted that composite resistance of different types of roughness equals to the sum of individual resistance (Shen and Li, 1973; Rauws, 1988; Weltz et al., 1992; Parsons et al., 1994; Hu and Abrahams, 2005). In a case where surface objects (e.g. stone or pebbles) and rainfall are the roughness elements in consideration, the following additive equation (also called
⁎ Fax: +1 510 885 2353. E-mail address: [email protected]. 0341-8162/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.catena.2009.03.010
fR&I = fR + fI
ð2Þ
where fR is the resistance when surface objects is the only roughness element and its density is R, fI is the resistance when rainfall is the only roughness element and its intensity is I, fR&I is the composite resistance when both roughness elements, with surface object density being R and rainfall intensity being I, are present in the flow. Linear superposition approach was initially proposed for river flows (Einstein and Banks, 1950; Yen, 2002) where roughness elements, such as rainfall impact to river surface and stones at the river bed, are normally separated from each other by the depth of flow. It may not be appropriate to use in overland flow studies because roughness elements commonly interfere with each other in shallow flows. For example, surface objects in overland flow often generate disturbance and turbulent vortex which extend vertically to the entire flow profile (Bunte and Poesen, 1993). When rainfall is added to the flow, raindrop impacts directly interfere with turbulent vertex, either amplify or suppressing them, causing the composite resistance likely to differ from the sum of surface object resistance and rainfall resistance. To investigate the interference between surface objects and rainfall in overland flow resistance, a sound study would 1) measure flow
G. Li / Catena 78 (2009) 154–158
resistance due to surface objects fR and the flow resistance due to rainfall impact fI separately; and 2) measure the composite resistance due to the presence of both rainfall and surface objects fR&I; and then 3) compare the sum of fR and fI and measured fR&I. However, flow resistance measured in lab or in field is almost always a composite resistance, often composed of surface resistance and form resistance or surface resistance and rainfall resistance etc. Unfortunately, there is currently not a method able to measure individual resistance of surface object fR or individual resistance of rainfall fI directly, making it impossible to investigate their interference and to evaluate Eq. (2) in its current form. It is therefore necessary to introduce friction factors fR′, fI′, and fR′&I′ where the magnitudes of their corresponding roughness elements are R′, I′, and R′&I′, to re-write Eq. (2) as: fR&I −ðfR V + fI VÞ = fR − fR V + fI − fI V which in turn, considering fR′&I′ = fR′ + fI′, can be written as ΔfR&I = ΔfR + ΔfI
ð3Þ
Where ΔfR&I = fR&I − fR′&I′ is the increment of composite resistance when both surface object density and rainfall intensity increase from R&I to R′&I′; ΔfR = fR − fR′ is the increment of flow resistance when surface object density increases from R to R′. ΔfI = fI − fI′ is the increment of rainfall resistance when rainfall intensity increases from I to I′. It should be noted that the determination of ΔfR does not actually require surface object to be the only roughness element in the flow. When rainfall is present while its intensity remains constant, the increase of surface objects from density R to R′ also results in ΔfR. Similarly, the determination of ΔfI does not require rainfall to be the only roughness element in the flow. When surface object is present while its density remains constant, the increase of rainfall intensity from I to I′ also results in ΔfI. Now, linear superposition approach and the additive equation (Eq. (2)) can be examined through Eq. (3) if direct measurements of ΔfR&I, ΔfR, and ΔfI can be made. Another important aspect of this superposition approach is in its linearity. For Eq. (2) to hold, the coefficients of independent variables need to be constant. This means that the importance of each individual resistance has to remain unchanged whatever the magnitude of other individual resistance is. This seems dubious. Li and Abrahams (1999) have shown that the role of rainfall in flow hydraulics and sediment transport processes changes with increasing flow power. It has also long been known that wave resistance, as
155
another example, diminishes when the impact of other roughness elements increases (Rouse, 1938). The linearity of the additive equation can be assessed by statistical regression as long as the experiment provides data that covers a wide range of flow conditions and diversified density of roughness elements. This study examines the validity of linear superposition approach when rainfall and surface roughness are considered in overland flow resistance. 2. Experimental setup and methods To test Eq. (3), a set of experiments were conducted in a 5.2 m long and 0.4 m wide flume with a smooth aluminum bed and Plexiglas walls (Fig. 1). It consisted of two parts: a lower part 3.6 m long and an upper steeper part 1.6 m long. For the experiments the lower part of the flume was inclined at a slope s of 2.0°. Water entered the upper part of the flume by overflowing from a head tank. The water inflow of 44.9 to 170.4 cm3 s− 1 was controlled by a gate valve and measured with a rotameter. Water discharge Q at the end of the flume was calculated by adding the rainfall onto the flume to the water inflow from the head tank. The density of the water ρ was assumed to be 1000 kg m− 3, while the kinematic viscosity ν was determined from water temperature which was 17 °C. The study focuses on two roughness elements: simulated rainfall and surface objects. PVC pipe connectors of 3.3 cm diameter, filled with concrete and sealed at both ends, were used as surface objects. They were randomly placed in the flume, perpendicular to the flume bed, and were tall enough to fully protrude above the water surface. The proportion of the bed covered by connectors R ranged from 0 to 20%. Rainfall was simulated using Spraco–Lechler full cone jet nozzles (Luk et al., 1986) mounted 0.3 m apart and 3.6 m above the center of the flume. Rainfall intensity I in experiment ranged from 0, 1.22, to 1.84 mm min− 1 (see the detailed experiment design in Table 1). The median drop sizes (Laws and Parsons, 1943) associated with these two intensities were 2.0 and 2.4 mm, respectively. Drop size increased with rainfall intensity because the spray cones intersected when more than one nozzles were operating, causing the water drops to collide and form larger drops. Given that the flow velocity from each nozzle exceeded 5 m s− 1, most drops reached terminal velocity. The spatial uniformity of the rainfall was evaluated using eight rain gages during nine 30-minute events. The coefficient of variation averaged 0.093, indicating a highly uniform distribution. The flow velocity u was determined at a 1.6 m long section of the lower end of the flume by a dye tracing method. The experiments were conducted in clear water flow with Reynolds number Re b 2000
Fig. 1. Flume set up.
156
G. Li / Catena 78 (2009) 154–158
Table 1 Experiment design. Surface object density R (%)
Rainfall intensity I (mm min− 1)
Measured friction factor f
14 (R2) 20 (R3) 7 (R1) 14 (R2) 20 (R3) 7 (R1) 14 (R2) 20 (R3)
0 (I0) 0 (I0) 1.22 (I1) 1.22 (I1) 1.22 (I1) 1.84 (I2) 1.84 (I2) 1.84 (I2)
fR2I0 fR3I0 fR1I1 fR2I1 fR3I1 fR1I2 fR2I2 fR3I2
(Re = 4q/ν, where q is unit flow discharge). The mean velocity was then obtained by multiplying a constant 0.67 to leading edge velocity (Horton et al., 1934). For each flow velocity, measurements were repeated 3 times from which two closest results were averaged to yield the mean value u. Mean flow depth d was then calculated using d = Q/uw and w = W(1 − R), where w is the mean flow width, W is the flume width, and R is the roughness element density which is determined from the total area covered by PVC pipe connectors divided by flume surface area. In total, 40 experiments were conducted and the details are listed in Table 2. Decades of study of flow hydraulics has shown that f varies with Re. To obtain Δf it is therefore essential to compare f values at the same Re. In current experiments, f was measured at slightly different Re (see Table 2) because it is impossible to precisely control the flow
discharge Q (and therefore the unit flow discharge q from which Re was determined) with a mechanic gate valve. To tackle the issue, the data was organized into five groups for which data fitting was performed to align comparable f to the same Re (see Fig. 2a–e). The method used straight lines to connect dots pair by pair on f−Re chart and then linearly interpolated f values to a given Re. The location of lines regarding Re is so chosen that only interpolation (no extrapolation) is involved. Δf of different roughness densities or intensities was then determined by comparing f values at given Re (see Table 3). Take Group a. as an example, ΔfR&I was calculated by comparing flow resistance fR2&I2 and fR1&I1when both surface object density and rainfall intensity increased from R1 (7%) and I1 (1.22 mm min− 1) to R2 (14%) and I2 (1.84 mm min− 1). ΔfR was determined by comparing flow resistance fR2&I1 and fR1&I1when surface object density increased from R1 to R2 while the rainfall intensity remained unchanged as I1. ΔfI was determined by comparing flow resistance fR1&I2 and fR1&I1 when rainfall intensity increased from I1 to I2 while the surface object density remained unchanged as R1. The comparison results are listed in Table 4. 3. Results Corresponding to the rationales stated in the Introduction, the experiment results were analyzed in two ways: (i) direct comparison of Δf to examine the validity of Eq. (3); and (ii) multiple regression of all involved resistance to investigate the possible variation of coefficients for each friction factor.
Table 2 Experiment results. Experiment number
Reynolds number Re
Froude number Fr
Base flow discharge Q (cm3 s− 1)
Roughness density R (%)
Rain intensity I (mm min− 1)
Mean flow velocity U (cm s− 1)
Friction factor f
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1084 1287 1490 1695 1899 1020 1220 1422 1626 1830 1008 1211 1416 1620 1824 1091 1292 1496 1700 1904 1022 1221 1423 1626 1830 1008 1211 1414 1619 1823 1098 1298 1501 1705 1909 1025 1223 1423 1626 1829
1.4 1.8 1.8 1.8 1.8 1.3 1.6 1.6 1.7 1.7 1.4 1.5 1.7 1.7 1.8 1.4 1.4 1.4 1.5 1.6 1.1 1.2 1.3 1.3 1.3 1.3 1.4 1.5 1.6 1.5 1.1 1.2 1.2 1.3 1.3 0.9 1.1 1.1 1.2 1.2
80.3 100.7 121.3 141.9 162.5 58.9 79.1 99.5 120.1 140.7 94.2 113.2 132.2 151.3 170.4 72.6 91.4 110.4 129.5 148.6 51.3 69.9 88.7 107.7 126.8 87.6 105.2 122.9 140.7 158.4 66.1 83.5 101.1 118.8 136.6 44.9 62.1 79.5 97.1 114.8
7 7 7 7 7 7 7 7 7 7 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84 0 0 0 0 0 1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84 0 0 0 0 0 1.22 1.22 1.22 1.22 1.22 1.84 1.84 1.84 1.84 1.84
18.2 22.1 23.3 24.6 25.2 16.6 20.4 21.2 22.8 24.4 17.7 19.1 22.1 23.1 24.6 17.4 19.1 20.0 21.9 23.3 14.5 16.9 18.2 19.9 20.6 16.5 18.3 20.2 22.1 22.6 15.1 16.9 18.3 19.7 20.8 13.0 15.3 17.0 18.2 18.6
3.50 2.31 2.28 2.19 2.29 4.30 2.78 2.88 2.65 2.45 3.51 3.34 2.54 2.56 2.36 3.99 3.57 3.60 3.14 2.91 6.51 4.92 4.59 4.02 4.05 4.35 3.81 3.31 2.90 3.07 6.18 5.23 4.72 4.34 4.10 9.05 6.60 5.60 5.25 5.47
G. Li / Catena 78 (2009) 154–158
157
Fig. 2. Illustrations of linear data fitting. Vertical axes represent the Darcy–Weisbach friction factor. Horizontal axes are Reynolds numbers. Dashed lines indicate the Reynolds numbers at which friction factor f is interpolated.
3.1. Direct comparison of Δf Five groups of a total of 20 comparisons between composite resistance and the sum of individual resistance were made. Values in columns ΔfR&I and ΔfR + ΔfI in Table 4 clearly demonstrated that ΔfR&I ≠ ΔfR + ΔfI :
ð4Þ
Simple t-test confirmed that ΔfR&I is significantly larger than ΔfR + ΔfI for all 16 comparisons in Groups a. to d. at 95% confidence level. The difference in percentage, indicated by ΔfR&I − (ΔfR + ΔfI) / (ΔfR + ΔfI), varies from 8% to 210%, averaging 47%. For Group e. result, t-test showed that there was no significant difference between ΔfR&I and ΔfR + ΔfI. Two of the four comparisons in Group e. showed that ΔfR&I was slightly less than ΔfR + ΔfI (the difference is −3% and −8%). Further examination of the Fig. 2 revealed that the result of Group e. was mainly influenced by one set of experiments where R = 14% and I = 1.22 mm Table 3 Determination of friction factor increments. Comparison group
ΔfR&I
ΔfR
ΔfI
a b c d e
fR2I2 − fR1I1 fR3I1 − fR2I0 fR3I2 − fR1I1 fR3I2 − fR2I0 fR3I2 − fR2I1
fR2I1 − fR1I1 fR3I0 − fR2I0 fR3I1 − fR1I1 fR3I0 − fR2I0 fR3I1 − fR2I1
fR1I2 − fR1I1 fR2I1 − fR2I0 fR1I2 − fR1I1 fR2I2 − fR2I0 fR2I2 − fR2I1
Note: ΔfR&I is the increment of composite resistance when both surface object density and rainfall intensity increase from R&I to R′&I′. ΔfR is the increment of surface object resistance when surface object density increases from R to R′ while rainfall intensity remains the same. ΔfI is the increment of rainfall resistance when rainfall intensity increases from I to I′ while surface object density remains the same.
min− 1. The measured fR2I1 in the experiments seemed to be lower than it should be (see Fig. 2b. and e.). This was likely due to a specific arrangement of surface objects in the flume (which was done randomly)
Table 4 Results of friction factor increments and their comparison. Reynolds number
ΔfR&I
ΔfR
Group a. 1091 1292 1496 1830 Group b. 1098 1298 1501 1823 Group c. 1098 1298 1501 1829 Group d. 1025 1223 1423 1823 Group e. 1098 1298 1501 1829
fR2I2 − fR1I1 2.48 2.49 2.10 1.79 fR3I1 − fR2I0 2.74 2.23 2.17 1.84 fR3I2 − fR1I1 4.73 3.92 3.19 3.21 fR3I2 − fR2I0 5.55 3.30 3.07 3.09 fR3I2 − fR2I1 4.14 2.64 1.86 2.47
fR2I1 − 0.52 1.26 1.32 0.73 fR3I0 − 0.68 0.60 0.60 0.71 fR3I1 − 2.76 2.93 2.44 1.93 fR3I0 − 0.81 0.47 0.83 0.71 fR3I1 − 2.20 1.66 1.13 1.20
ΔfI fR1I1
fR2I0
fR1I1
fR2I0
fR2I1
fR1I2 − 0.28 0.51 0.52 0.19 fR2I1 − 0.54 0.58 1.04 0.65 fR1I2 − 0.29 0.51 0.51 0.19 fR2I2 − 2.99 1.61 1.56 1.68 fR2I2 − 1.91 1.22 0.79 1.05
ΔfR + ΔfI
ΔfR&I −ðΔfR + ΔfI Þ ðΔfR + ΔfI Þ
0.80 1.77 1.84 0.92
210% 41% 14% 95%
1.22 1.18 1.64 1.36
125% 89% 32% 35%
3.05 3.44 2.95 2.12
55% 14% 8% 51%
3.8 2.08 2.39 2.39
46% 59% 28% 29%
4.11 2.88 1.92 2.25
1% − 8% − 3% 10%
fR1I1
fR2I0
fR1I1
fR2I0
fR2I1
Note: Detailed flow condition of specific f, denoted by its subscript, can be found in Table 1. The Reynolds number in each group corresponds to dashed lines in Fig. 2.
158
G. Li / Catena 78 (2009) 154–158
that happened to assemble a less resistance and promoted a faster water flow. 3.2. Multiple regression analysis To investigate whether the role of individual resistance changes across the range of experiment conditions, a multiple regression was performed with f as dependent variable, R, I, and the product of both R ⁎ I as well as Re as independent variables. Initial regression yielded: 0:873
or
− 0:867
f = ð273:4R + 758:3I + 98:7RTIÞ Re 0:873 − 0:867 f = ð273:4R + ð758:3 + 98:7RÞI Þ Re
ð5Þ
with residual corrected R2 = 0.934 and N = 40. The result confirmed, as expected, that f increases with R and I and decreases with Re. It seems that f also increases with the term R ⁎ I. If so, the coefficient for independent variable I would be (758.3 + 98.7R) which is not a constant but varies with the value of R (the same is true for the coefficient of independent variable R). This would have concluded that the contribution of I to f depends on R (or vice versa). However, a significance test showed that the inclusion of the term R ⁎ I was statistically insignificant and improved the R2 only by 0.9%. Power coefficient for Re (− 0.867) was not statistically different from − 1 either and improved the R2 merely by 0.6%. This agreed with Moody diagram in which f varies with Re− 1 for Re b 2000. A revised regression without R ⁎ I and with power coefficient for Re as − 1 yielded: 1:204
f = ð59:0R + 343:8IÞ
Re
−1
ð6Þ
2
with residual corrected R = 0.916 and N = 40. Significance test showed that R, I and Re were all statistically significant. 4. Conclusion and discussion The current experiments were conducted in a smooth-bed flume of 2° slope. Reynolds number Re was less than 2000. Two roughness elements were investigated for their possible interference in overland flow resistance: surface objects and rainfall. The result shows that surface objects (and their turbulent vortex) interfere with raindrop impact to the flow which makes the composite resistance differ to the sum of the two individual resistances. The interference may restrain or strengthen friction factor on one another, leading the composite resistance to decrease or to increase. For river flows, Vanoni and Nomicos (1959) had published a similar result which showed that composite resistance decreased by about 5 to 28% when sediment load was introduced to turbulent channel flow. They ascribed the decreasing resistance to sediment load suppressing flow turbulence. Current experiment also tried to explore whether the interference of individual resistance increases or decreases composite resistance. The analysis did show that in most cases ΔfR&I N ΔfR + ΔfI. Great care must be exercised here not to conclude that fR&I N fR + fI though. It would require establishment of other conditions that are unknown at this point (e.g. the linear relationship between fR&I and R&I, fR and R, fI and I). Further study with extended flow condition and diversified roughness elements is needed to reveal the full spectrum of the relationship in this matter. Other studies suggested that the contribution of each roughness element to flow resistance may not be independent of other roughness elements (e.g. rough elements generate wakes that reduce grain resistance, Morris, 1955; Abrahams and Parsons, 1991) and the coefficients of independent friction factors in the additive equation may not be constant (e.g. rainfall resistance declines with Re, Shen and Li, 1973). However, current experiment and analysis did not reveal the changing nature of rainfall in its contribution to flow resistance with varying density of surface objects (or vice versa). This was most
likely due to a relatively narrow range of roughness density covered in the experiments. It is suggested to further investigate in the future the possible diminishing role of a roughness element (e.g. rainfall) in overland flow resistance due to increased magnitude of other roughness element (e.g. surface objects). Finally, it is worth pointing out that roll waves can easily develop in flume and in field experiments (Benjamin, 1957; Yoon and Wenzel, 1971; Emmett, 1978;) especially when flow is barely impacted by roughness elements (Rouse, 1938) or when Froude number Fr, defined as u/(gd)0.5, is greater than 2 (Brock, 1970; Liggett, 1975; Liu et al., 2005). Roll wave is a form of energy expenditure and adds another resistance to the flow. In order not to complicate the analysis, the current study did not include the experiments with visually identifiable roll waves. Acknowledgements The author greatly appreciates the support of Professor Sean Bennett, Department of Geography, State University of New York at Buffalo and Professor Shixiong Hu, Department of Geography, East Stroudbury University of Pennsylvania for their support and assistance during the experiment conducted in the Geomorphology Lab at State University of New York at Buffalo. The author also thanks Professor Michael Woldenberg and Professor Athol Abrahams, Department of Geography, State University of New York at Buffalo for their valuable comments and suggestions. References Abrahams, A.D., Parsons, A.J., 1991. Resistance to overland flow on desert pavement and its implications for sediment transport modeling. Water Resour. Res. 27, 1827–1836. Abrahams, A.D., Li, G., 1998. Effect of saltating sediment on flow resistance and bed roughness in overland flow. Earth Surf. Process. Landf. 23, 953–960. Benjamin, T.B., 1957. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574. Brock, R.R., 1970. Periodic permanent roll waves. J. Hydraul. Div. Am. Soc. Civ. Eng. 95, 1401–1428. Bunte, K., Poesen, J., 1993. Effects of rock fragment covers on erosion and transport of noncohesive sediment by shallow overland flow. Water Resour. Res. 29, 1415–1424. Einstein, H.A., Banks, R.B., 1950. Fluid resistance of composite roughness. Trans. Am. Geophys. Union 31 (4), 603–610. Emmett, W.W., 1978. Overland flow. In: Kirkby, M.J. (Ed.), Hillslope Hydrology. Wilely, New York, pp. 145–175. Horton, R.E., Leach, H.R., Van Vliet, R., 1934. Laminar sheet-flow. Trans. Am. Geophys. Union 15, 393–404. Hu, S., Abrahams, A.D., 2005. The effect of bed mobility on resistance to overland flow. Earth Surf. Process. Landf. 30, 1461–1470. Laws, J.O., Parsons, D.A., 1943. The relation of raindrop size to intensity. Trans. Am. Geophys. Union 24, 452–460. Li, G., Abrahams, A.D., 1999. Controls of sediment transport capacity in laminar interrill flow on stone-covered surfaces. Water Resour. Res. 35, 305–310. Liggett, J.A., 1975. Unsteady Flow in Open Channels. Water Resources Publications, Fort Collins, Colorado. Liu, Q.Q., Chen, L., Li, J.C., Singh, V.P., 2005. Roll waves in overland flow. J. Hydrol. Eng. 10 (2), 110–117. Luk, S.H., Abrahams, A.D., Parsons, A.J., 1986. A simple rainfall simulator and trickle system for hydro-geomorphological experiments. Phys. Geogr. 7, 344–356. Morris, H.M., 1955. Flow in rough conduits. Trans. Am. Soc. Civ. Eng. 120, 373–410. Parsons, A.J., Abrahams, A.D., Wainwright, J., 1994. On determining resistance to interrill overland flow. Water Resour. Res. 30, 3515–3521. Rauws, G., 1988. Laboratory experiments on resistance to overland flow due to composite roughness. J. Hydrol. 103, 37–52. Rouse, H., 1938. Fluid Mechanics for Hydraulic Engineers. McGraw-Hill, New York. Shen, H.W., Li, R.-M., 1973. Rainfall effect on sheet flow over smooth surface. J. Hydraul. Div. Am. Soc. Civ. Eng. 99, 771–792. Smith, M.W., Cox, N.J., Bracken, L.J., 2007. Applying flow resistance equations to overland flows. Prog. Phys. Geogr. 31 (4), 363–387. Vanoni, V.A., Nomicos, G., 1959. Resistance properties of sediment-laden streams. J. Hydraul. Div. Am. Soc. Civ. Eng. 85 (5), 77–107. Weltz, M.A., Arslan, A.B., Lane, L.J., 1992. Hydraulic roughness coefficient for native rangelands. J. Irrig. Drain. Eng. 118 (5), 776–790. Yen, B.C., 2002. Open channel flow resistance. J. Hydraul. Eng. 128 (1), 20–39. Yoon, Y.N., Wenzel Jr., H.G., 1971. Mechanics of sheet flow under simulated rainfall. J. Hydraul. Div. Am. Soc. Civ. Eng. 97, 1367–1386.